I count 14 nights, assuming you land on the 21st.
When you land in Rome, proceed immediately to Florence (assuming your flight arrives fairly early?) by train. Florence is about art. If you want to see the big museums, reserve in advance and spend at least 3 nights here. Look at what you can actually do in a day without exhausting yourselves in the heat. We find one museum or other “big site” in the morning and one more minor site or activity in the afternoon is about the right pace, with a rest before dinner.
From Florence, the Cinque Terre is reachable easily enough but it still takes awhile so by the time you get there and settle in, most of the day is gone. If you really want to see the villages and take a hike or two, 3 nights is the minimum.
Saving 4 nights *minimum * for Rome at the end, you have 5 nights to spend, so what interests you? If Venice, go there before Florence. Milan does not have the great sites of other cities so probably skippable on a first trip. Again, 3 nights minimum in Venice. But Lake Como is also pretty much a 3 night location, IMO, so maybe stick with 4 locations in total, like this:
- 4 Florence with a day trip to Siena
- 3 Cinque Terre
- 3 Lake Como
- 5 Rome with perhaps a day trip to Ostia Antica or Villa d’Este in Tivoli
Do you have Rick Steves’ Italy guide book? Essential reading for logistics of all these places.